Solutions for certain rectangular slabs continuous over flexible ...

SOLUTIONS FOR CERTAIN RECTANGULAR
SLABS
wo, given by (219), there may then be substituted the deflection of
the infinitely long slab of span b, simply supported along the edges
x = + b/2, and carrying a concentrated load at the center of the span.
The supports at y = ±a/2 are moved sufficiently far from the load
by making a = 27rb in Equations (221) and (222).
Let Moz and Mo, designate the bending moments under the load
when it is at the center of the span of an infinitely long slab having
simply supported edges at x = + b/2. Let Mox and MO', represent
similar moments when the edges of the slab at x = + b/2 are fixed.
Let Mox and Mo, represent similar moments when the slab extends a
considerable distance in the directions of positive and negative x
and is continuous over non-deflecting supports at x = ±b/2. Under
these conditions, for Poisson's ratio p = 0.15, the moments under
the load are
with simply
Mox = Mox
supported edges M 0 o = Mox - 0.0676P
! M0• = Mox - 0.0699P
with fixed edges - 0.0(227)
M', = Mo, - 0.03863P = Mo - 0.1063P
with continuity M0' = M 0 x - 0.0293P (228)
of slab M"' = Mo, - 0.01905P = Moý - 0.0867P.
The numerical value of the corrections to Mo 0 and Mo, given in (228)
were computed with a = 2irb in the corresponding correction functions
(225). The initial quantities required in (226) may be obtained
from (18) or from the numerical values and curves given by Westergaard.
The corrections to Mo 0 and Moy noted in (227) were given
by Westergaard* and were verified from (225) with a = 27rb and
with k5 substituted for ks.
It may be observed from (227) and (228) that the correction to
M 0 o due to continuity of the slab is 42 per cent of the correction due
to fixing the slab. Similarly it may be observed that the correction
to M 0 o, due to continuity is 49 per cent of the correction due to fixing
the slab. It is apparent, therefore, that the continuous slab is not,
in this instance, one having a definite "percentage of fixity," but
may, nevertheless, be treated as "50 per cent fixed" with a satisfactory
*H. M. Westergaard, Computation of Stresses in Bridge Slabs Due to Concentrated Loads, Public
Roads, V. 11, No. 1, March, 1930, p. 20; see equations 104 and 105.